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<Paper uid="C88-2107">
  <Title>REFERENCES</Title>
  <Section position="2" start_page="0" end_page="0" type="metho">
    <SectionTitle>
2. SYSTEM OVERVIEW
</SectionTitle>
    <Paragraph position="0"> Me have designed an exporimental instruction system in mathematics. The system operates in a limited domain: it is capable of solving a restricted class of stoc~ problems in algebra EOC Ssoondar~ Schools in Bulgaria (the so ~alled &amp;quot;number problems&amp;quot;). The system accepts non-stylized stor'y problems in Bulgarian as they can be Found in mathematical textbooks of are spontaneously Formulated bU the user. It solves the probl~m, and is capable of providing either of the Following 3 options:  (a~ Result (resultant number(s) are displayed), (b9 Equations (the equation(s) to which problems translate are displayed). (c) Paraphrases (one or more  &amp;quot;hinting&amp;quot; paraphrases are displayed, together with the text of the orioinal problem).</Paragraph>
    <Paragraph position="1"> All the three options serve as CdiFFecent degrees of) hinting needed in case the users (Secondary 5chool pupils) have probl~ms with Finding a solution, Furtheron, we Focus on problems concerned with the hinting paraphrastic Facility oF the system.</Paragraph>
  </Section>
  <Section position="3" start_page="0" end_page="507" type="metho">
    <SectionTitle>
3, GENERAL REQUIREMENTS TO A
HINTING PARAPHRASE
</SectionTitle>
    <Paragraph position="0"> The profit of using a paraphrase, or a &amp;quot;reO'oreulation&amp;quot;, oF a problem as a , heuristic tool has been emphasized b U researchers in heuristics, pedagogy and psy~hmlogy of education. Nevertheless, such a possibility is usually be qond the s~ope OF i~letruction systems (51eeman, Brush laB2, Weischedsl st el IS'TO, PLJlrnaN 19B~), l he question still remains as to what can count as a hinting pararhPSasa (HP) (obviously , not ~ paraphrase can serve this purpose equally well). Has*no ourselves on research in mathematical pedagogy and psgchol inguistics (since conceptual and linguistic structures in this earl, g age are known to be stronoIN  interdependent\], we derived the followlno general requirammnts to a HP: i. The HP should ~ the original problem (OP) as rsoards the users of the system ( by this we msan simplification Of OP in both conceptual and linguistic aspects with respect to the task assigned, viz. to fomalise (-translate into equations) th~ OP.</Paragraph>
    <Paragraph position="1">  2. The HP should be ~. From the OP (this requirement is self-evident). 3. The HP should k_eeep close to the OP  From a conceptual and linguistic viewpoi~lts (this is to ensure that the USers conceive the &amp;quot;sameness&amp;quot; of HP and OF).</Paragraph>
    <Paragraph position="2"> Since the most important task of th~ HP is to simplif~ the translation of verbally formulated problems into equations (solving these equations being as a rule wnproblematic for childrHn), ws took the Following ~sneral solution regardin 0 an &amp;quot;appropriats&amp;quot; HP: An &amp;quot;appropriate&amp;quot; HF to a problem is the one that can be, somewhat metaphorically expressed, literallu translated i~to the respmctive equations of this problem.</Paragraph>
    <Paragraph position="3"> Obviously, this would, to the greatest extent possible, simplify the OP (in the sense in which in the translation from one NL to another, the easiest to perform is the literal translation\]. This de~ision is further supported by the faot that pupils usually t~ans\].ate to themselves the OP into intermediary languaoe which is most close to the equations derivable from this problem.</Paragraph>
  </Section>
  <Section position="4" start_page="507" end_page="509" type="metho">
    <SectionTitle>
~. CONSTRAINTS ON
</SectionTitle>
    <Paragraph position="0"> &amp;quot;APPROPRIATE&amp;quot; HPs From what is stated above, a number of specific constraints on the content and form of the HP can be derived. We briefly mention them below in connection with two of the major decisions that have to be made in a oeneration presses: first, makin 0 a decisions as to the structure of the HP (i.o. determinin 0 what and whsn to sag, or an ordered message to be conveyed), and, secondly, making a decision as to the y~rbal formulatiom of the discourse structure of the HP (i.e.</Paragraph>
    <Paragraph position="1"> determinin 0 how to express this in/ormation in Huloarian, what syntantic structures to use, what lexemes, etc.).</Paragraph>
    <Paragraph position="2"> At the first staos, we should gain in conceptual, and, at the seoond stage, in linouistic simplification, thus approximating the requirement as to the litsralness we have imposed.</Paragraph>
    <Paragraph position="3"> ~.i. Discourse structure In the light oF our aims, it is clear that the discourse structure o~ the HPs should be standardized, or ~, This means that we need not bs concerned (like most scholars workin~ on discourse organization, e.g. Mann 198~, McKswon  IH853 with ~\[W~typss of discourse structures of actual texts in the domain oE interest, but rather with ~ discourse pattern that satisfies the discourse ooal.</Paragraph>
    <Paragraph position="4"> Each of the texts in our domain, story problems in aloebra For Secondary 5chools, is known to be characterized h u Wi!knownf~!, (i.e. what is looked for in the problem), and ~ (i.e. the equation(s\], relating the unknown(s\], or variables, to the givenfs), or constants, in the problem). Some problems also involve ~uxiliaru~ (i0e. further unknown(s), often mentioned in the problem formulation somewhat misleadingly (e.g. &amp;quot;...Another number is ~.~&amp;quot;\], which have to be manipwlatad~ but are not themselves part of the solution\].</Paragraph>
    <Paragraph position="5"> Yhs discour58 structure of the HPs, thsreffore, will have to reflect the basic conoeptual constituents of the prohlems:  I. the unknown(s\] 2. the auxiliary unknownCs\] &lt;optionally&gt; 3. the condition(s),  in bb~ partiowl~r order.</Paragraph>
    <Paragraph position="6"> It may be noted that a lot cf problems, as they are formulated in mathematics textbooks, do not actually satisfy this discourse sohsma: the unknowns are interspersed in the text, the unknownfs) and auxiliary unknown(s\] ar~ not sxplioitly discrimisated, the conditions precede (auxiliary\] unknown(s), etc.</Paragraph>
    <Paragraph position="7"> For instanoe, a typical problem to be found in a textbook maw begin as follows: &amp;quot;The sum of two numbers is B...&amp;quot; Clearlu, 5tartin 0 the problem formulation hU a condition, instead of with declaring first the unknown(s), is misleading. Thus, notice that this problem ma~ have quite different continuations, among which ...The first number is 2, Which is the second? in which we have just ng~q~_ unknown, or ...Their product is 12.</Paragraph>
    <Paragraph position="8"> What are these numbers? a version in which there arm two urlknowns. The resolution of this local ambiguity requires additional i~itallectual effort on the part of the pupilj ~o.readJno, etc., ~ircumstances which ou~ HPs should evade.</Paragraph>
    <Paragraph position="9"> In addition to dssoribino the major' conceptual constituents of the problems~ in the canonical discourse structuma of the HPs, the monditions of problems themselves, usuall~ compound propositio|is, should be brok@n_~o n arts. The ordering of these propositions should  (!) It&amp;quot; th~ ~Um Or&amp;quot; aria numbe~ ~iith ('AT) mhl~h i&lt;&lt;~ mith t:- ~tlIi~\].17t\[&amp;quot; 4\[1}1~ill it (3) J.~ lilultlpliPS-Id i\]~ ~.i ('i) ~j~'Ju l, li}_.! L:=lnd &amp;quot;h|ic~ p~-c~duct u17 t'.h~t ~4~j~l;t~'tlJ i-ilJirih~3~- mit~t thb\] lltllll!lKJ~.' ~\]. (;5) b'irl.rI th\[~ t#.!,c!~t llUllrlt:l~?F, (i-L) ~li~lJtli~r nlJmhur i5~ \[L!v~tio (\['1) ~IJLt th~ tbio IlLliflhEi\['t~.  ~\[~l \[:umpn~'isei~ ~ith t|lS \[JP, tlis HP ~xplluh%tSS th~ tt.o rlumb~r5 of the pF\[Jblem that mill bs PS~Ll~'t\]i~m manipulstsd: Fi~:st, t  |i~-~ u~ikno ,,~11, mi-ld ~ thatl, th~ aux i i +- arH lJrlkrlol~n. In clause (13 of t|iS lip ths ~p~atioe off ~dditio~ is imp\].icitlu f~iv~n b U its ~-~sult (&amp;quot;ths ~um&amp;quot;), mhr~as iH clause (39 F~F t\]lO HP th~ same upeF~tio~ J.s ~laborated h H arl axpli~it msntlorlino ~Jf thPS~ paFticula~ ~ avithmstic~l npeFation ~Jf nddltiun ~ The imhsddsd relative clause (2{3 ~)i:&amp;quot; tlls ~P is sxprosssd sepaFat~\]iu \[:ram  the lii\[~i~ sentence in the HP C(53 arid (7) el|? 'h\]'ll:~ \]|P). This pFovidss a puSsJ.bi.LitL4, ~,&amp;quot;~adi:I\[~ t}1o ~ndition OC the pFoblem |;'rOiil Isft ta ciEIht, tn mFite damn, sequsntialJ.u ~nd :i. ~ldl~pr~Tldsnt 1U, hhs diffff\[~rent oquatiuns, in the paraphrase ~ff th.~.</Paragraph>
    <Paragraph position="10"> ~-~lativo clause (2) of the \[\]P, the zdeg81ation '&amp;quot;is sili~lla~ than&amp;quot;, knumll to 138 ~i\]nEusino ffe~,&amp;quot; small child~'en, is ~-splacod hi\] its ~ov~sspOild i rllJ operat ioi~ ~'subtra~tlon&amp;quot;, and tl-ls pFOrlLmlir\]a l FSffBFSIJUI~ (sXpFSsssd in the Eel/fish test \[,~ith &amp;quot;it&amp;quot;3 is avsidsd0 Notice also that ('i) EFOm t|'is \[JP and (5) if,Fern tile HP ar~ ~ihvas~d in th~ ~ame ~ta U (thus pFes~Fvin\[~ vavtlal Samaesss off the OP and tb~ HP), The csoFOallised text of the HP ~an hs s~sl\] to Si~lllfioantlu simpliffu th~ DP  equations obtained as a ~esult of the pa~si~S phase fie the sustem is in a &amp;quot;~ssult&amp;quot; muds).</Paragraph>
    <Paragraph position="11"> ?he gsns~atiun process ~uss thL'ou~h ti.~ major phases. The pa~aphra~ti~; facilit U of the sustsm has t~o ~ompun~nts, vssponsihls ~o~ the tasks at these phaes~: the ~, add the ~X.</Paragraph>
    <Paragraph position="12"> In the first phase, the Canonizsr.</Paragraph>
    <Paragraph position="13"> ~onstruuts the disoouFs8 stFucture, of thE~ canonical fD~m, sff the HP. Ths pFu~sss includes the reprsssstatisn sff the ' dis~ouvsQ st~'uctur'e into a ssqusncs o~&amp;quot; slsmsntar.~ pr-upositiens, +-nstaritisted b~  the r~sult dmrivsd bg the Analysis module. This sequsnom b~gins with the proposition describing the unknown(s), and, optiooall~, propositions for auxiliar~ u~kr~ownfs). In the sequence follow the propositions desc~-ibing conditions C~squations).</Paragraph>
    <Paragraph position="14"> For example, as a ~'~sult o~ th~ analysis of the OF, mentioned in Sscto 5, the following sequence As obtained:</Paragraph>
    <Paragraph position="16"> The Canonizer shifts the last proposition unknoen(X) at the begining of the sequence of propositions sod adds get another proposition auxiliar~ unkno~m(Y)0 As a result</Paragraph>
    <Paragraph position="18"> is obtained.</Paragraph>
    <Paragraph position="19"> Each compound proposition of the latter type is substituted with an equivalent ~ ~. In order to achieve this, all oonstitusnt propositions are substituted bu variablss~ after&amp;quot; which the simple proposition obtained is unified with the compound proposition.</Paragraph>
    <Paragraph position="20"> In the above case, Erom the unification of the two compound propositions &amp;quot;equal&amp;quot; wlth the simple proposition equal(~,B), we obtain:</Paragraph>
    <Paragraph position="22"> where the expressions in braces are substitutions.</Paragraph>
    <Paragraph position="23"> The propositional expression thus describes the ~ of obtaining the compound proposition in question From simple propositions.</Paragraph>
    <Paragraph position="24"> AFter the substitution of Bach compound proposition of the equivalent propositional expression, the folloeing  The canonical representation used is easilg seen to have certain advantages. On the one hand, it explicates all computations nsoessarg ~or construction o~ the sostem of equations, and, on the other hand, it defines a l~i_~erballzati_o!l, to be used b~ the Generator, in which, First of all, the simple propositions are verbalized, then their verbalizations are  used in the verbalization of the compound propositions at the next higher level of hisrav~hu~ and so on0 The text to be obtained ~ollowing such a plan of verbalization can be literall u translated into a system cf equations b U virtue of the Fact that the text itself is ~snerat~d in inverse order&amp;quot; - From simple to compound proposihions, In the second phase of the process of generation of the HFs, the oanonical Fo~m of the HPs is translated into Bulgarian text bH the Generator. The 8enerato~ o itself is a.~!l~ &amp;quot; . &amp;quot; ~_~ m~ (the templates used Fo~ generation)deg Each template describes a sgntacti~ construction by means o~ particular wordForms, lexi~al classes and variables. Some of the templates are used &amp;quot;tO propagate anaphorioal relations Cdsfinite NPs, or pronominal references).</Paragraph>
    <Paragraph position="25"> As already mentionod, the Generator follows the plan For verbalization defined bg the canonlcal representation. ~ set of s Is&amp;quot; ' governs the choice of particular templates, ~L~uniFicatlon begins. In case of alternatives as to the choice OF a template, the Generator consults the derivational historg of the analysis, which is kept in a special register, and selects the template, and the concrete verbal ~o~mulation, used in the OP (this 8nsuring partial &amp;quot;sameness&amp;quot; of HPs and OPs).</Paragraph>
    <Paragraph position="26"> Yhe system is implemented in PROLO~-2 and runs on IBM RTs and compatibles.</Paragraph>
  </Section>
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